Preprint 7/2008

Compensated Compactness, Separately convex Functions and Interpolatory Estimates between Riesz Transforms and Haar Projections

Jihoon Lee, Paul F. X. Müller, and Stefan Müller

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Submission date: 31. Jan. 2008
Pages: 52
published in: Communications in partial differential equations, 36 (2011) 4, p. 547-601 
DOI number (of the published article): 10.1080/03605301003793382
MSC-Numbers: 49J45, 42C15, 35B35
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In this work we prove sharp interpolatory estimates that exhibit a new link between Riesz transforms and directional projections of the Haar system in formula12 To a given direction formula14 we let formula16 be the orthogonal projection onto the span of those Haar functions that oscillate along the coordinates formula18 When formula20 the identity operator and the Riesz transform formula22 provide a logarithmically convex estimate for the formula24 norm of formula26 see Theorem 1.1. Apart from its intrinsic interest Theorem 1.1 has direct applications to variational integrals, the theory of compensated compactness, Young measures, and to the relation between rank one and quasi convex functions. In particular we exploit our Theorem 1.1 in the course of proving a conjecture of L. Tartar on semi-continuity of separately convex integrands; see Theorem 1.5.

18.10.2019, 02:13